Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / M / 3 / c

A linear combination of independent normal random variables is normal. Its mean is $2\cdot 0+0=0$, and independence removes the cross-covariance term in its variance:

The identity $Y=aX+W$ forces $b=1-2a$ and $c=1-a$. Independence will follow from zero covariance, because $X,W$ are jointly Gaussian random variables. That covariance isThusThese values give $W=(-Z_1+2Z_2)/5$ with variance $1/5$ and zero covariance with $X$. Joint normality then proves the requested independence.

From the preceding decomposition, $Y=(3/5)X+W$, where $W$ is independent of $X$ and has mean zero. Taking conditional expectation gives $\mathbb{E}[Y\mid X]=(3/5)X$. The corresponding continuous version at $X=5$ is